 Solution

Let $e_j=x-x^{(j)}$. The underlying Richardson iteration gives $e_j=(I-A{)}^j{e}_0$. The polynomials $1,(1-t),\dots ,(1-t{)}^k$ form a basis for polynomials of degree at most $k$. Thus writeDefine $y^{(k)}={\sum }_{j=0}^k{c}_j{x}^{(j)}$. Since the coefficients sum to one,The spectral theorem for real symmetric matrices supplies an orthonormal basis of eigenvectors. In that basis $p_k(A)$ has diagonal entries $p_k({\lambda }_i)$, soThe algebraic recombination is valid in exact arithmetic even if the original unaccelerated iterates themselves fail to converge.